Problem 3

Part of 2008 VJIMC

Problems(2)

f^(n+1)(x)>f^(n)(x)+c, f analytic

Source: VJIMC 2008 1.3

c\in\mathbb R

for which there exists an infinitely differentiable function

f:\mathbb R\to\mathbb R

such that for all

n\in\mathbb N

x\in\mathbb R

f^{(n+1)}(x)>f^{(n)}(x)+c.

differential equationcalculus

1,sqrt[n]{n},sqrt[m]{m} linearly dependent over Q

Source: VJIMC 2008 2.3

Find all pairs of natural numbers

(n,m)

1<n<m

such that the numbers

1

\sqrt[n]n

\sqrt[m]m

are linearly dependent over the field of rational numbers

\mathbb Q